We report the discovery of an extremely large (R_b ~ 2"77 ~ 4.2 kpc) core in the brightest cluster galaxy, IC 1101, of the rich galaxy cluster Abell 2029. Luminous core-S\'ersic galaxies contain depleted cores---with sizes (R_b) typically 20 - 500 pc---that are thought to be formed by coalescing black hole binaries. We fit a (double nucleus) + (spheroid) + (intermediate-scale component) + (stellar halo) model to the HST surface brightness profile of IC 1101, finding the largest core size measured in any galaxy to date. This core is an order of magnitude larger than those typically measured for core-S\'ersic galaxies. We find that the spheroid's V-band absolute magnitude (M_V) of -23.8 mag (~ 25% of the total galaxy light, i.e., including the stellar halo) is faint for the large R_b, such that the observed core is 1.02 dex ~ 3.4 sigma_s (rms scatter) larger than that estimated from the R_ b -M_V relation. The suspected scouring process has produced a large stellar mass deficit (M_def) ~ 4.9 X 10^11 M_sun, i.e., a luminosity deficit ~ 28% of the spheroid's luminosity prior to the depletion. Using IC 1101's black hole mass (M_BH) estimated from the M_BH-σ, M_BH-L and M_BH-M_* relations, we measure an excessive and unrealistically high number of dry major mergers for IC 1101 (i.e., $\mathcal{N} \ga 76$) as traced by the large M_def/M_ BH ratios of 38-101. The large core, high mass deficit and oversized M_def/M_ BH ratio of IC 1101 suggest that the depleted core was scoured by overmassive SMBH binaries with a final coalesced mass M_BH ~ (4 -10) X 10^10 M_sun, i.e., ~ (1.7- 3.2) X sigma_s larger than the black hole masses estimated using the spheroid's σ, L and M_*. The large core might be partly due to oscillatory core passages by a gravitational radiation-recoiled black hole.